{
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  {
   "cell_type": "markdown",
   "id": "73bd968b-d970-4a05-94ef-4e7abf990827",
   "metadata": {},
   "source": [
    "Chapter 02\n",
    "\n",
    "# 向量内积\n",
    "Book_4《矩阵力量》 | 鸢尾花书：从加减乘除到机器学习 (第二版)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "90d57742-54b5-4967-9de6-2b97f3ff13ce",
   "metadata": {},
   "source": [
    "该代码定义了两个二维向量 $a$ 和 $b$，并计算它们的内积。向量 $a$ 和 $b$ 的定义分别为：\n",
    "\n",
    "$$\n",
    "a = \\begin{bmatrix} 4 & 3 \\end{bmatrix}, \\quad b = \\begin{bmatrix} 5 & -2 \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "代码首先通过 `np.inner` 函数计算行向量的内积，其计算公式为：\n",
    "\n",
    "$$\n",
    "a \\cdot b = 4 \\cdot 5 + 3 \\cdot (-2) = 20 - 6 = 14\n",
    "$$\n",
    "\n",
    "接着，代码将 $a$ 和 $b$ 定义为列向量形式 $a_2$ 和 $b_2$，并通过矩阵转置与矩阵乘法计算内积：\n",
    "\n",
    "$$\n",
    "a_2^T \\cdot b_2 = \\begin{bmatrix} 4 & 3 \\end{bmatrix} \\cdot \\begin{bmatrix} 5 \\\\ -2 \\end{bmatrix} = 14\n",
    "$$\n",
    "\n",
    "该过程展示了内积计算的不同实现方式，包括使用 `np.inner` 和转置矩阵乘法。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a869bcd3-b5b0-48a6-8473-e1af9efa124c",
   "metadata": {},
   "source": [
    "## 导入所需库"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "52854f8c-f415-452c-8b0f-ba216cedb4a3",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np  # 导入NumPy库，用于数值计算"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a895dedd-943b-4b5e-a9d8-6c958e1ec18e",
   "metadata": {},
   "source": [
    "## 定义两个行向量"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "13d6cc5a-824f-4537-8d80-5e4a0c6146a7",
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.array([[4, 3]])  # 定义向量a，值为[4, 3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "89d1799e-2c61-4410-83fd-401db971b8fd",
   "metadata": {},
   "outputs": [],
   "source": [
    "b = np.array([[5, -2]])  # 定义向量b，值为[5, -2]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3277e077-e975-4354-b79a-5bc9e77d5666",
   "metadata": {},
   "source": [
    "## 计算内积"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "e4008c0e-94ff-4cb5-8139-432ed0048692",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[14]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_dot_b = np.inner(a, b)  # 使用np.inner计算a和b的内积\n",
    "a_dot_b"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b6a1c7e0-b5e4-4303-b2a8-19e7a0d72cbb",
   "metadata": {},
   "source": [
    "## 定义两个列向量"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c30e546b-1f82-43c7-b4d5-163394f406c3",
   "metadata": {},
   "outputs": [],
   "source": [
    "a_2 = np.array([[4], [3]])  # 定义列向量a_2，值为[4, 3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "41ee49f6-1d62-4f0e-895e-671ae32ad301",
   "metadata": {},
   "outputs": [],
   "source": [
    "b_2 = np.array([[5], [-2]])  # 定义列向量b_2，值为[5, -2]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d6a74fd1-5de1-486c-98fb-ff21a16d5158",
   "metadata": {},
   "source": [
    "## 计算转置后内积"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "ce1247f9-be4e-40aa-be27-9b9b4d8f9b03",
   "metadata": {},
   "outputs": [],
   "source": [
    "a_dot_b_2 = a_2.T @ b_2  # 使用矩阵乘法计算a_2转置和b_2的内积"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "85a80909-2aac-49ed-bb7a-f8cc6b80ee7d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[14]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_dot_b_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ecd322f4-f919-4be2-adc3-69d28ef25e69",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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